The control signal of a fixed gain PID controller (presented in the link below) can be written as
y(t) = Kp u(t) + Ki int{u(t)} + Kd(t) d/dt{u(t)}. Note in the figure that, the error signal is been multiplied by the integral gain (Ki), before the integral. Considering the time invariant controller, does not matter if the gain is before or after the integration.
Now suppose that, the gains are time varying, and for simplicity the Kd(t) = 0 for all t>0, then the signal control can be written as
y(t) = Kp(t) u(t) + int{Ki(t)u(t)}, if the gain is before the integration, and
y(t) = Kp(t) u(t) + Ki(t) int{u(t)}, otherwise.
Due the fact of the gains been time varying, this is a nonlinear controller, and there are clear difference between these implementations (the part of signal composed by the integration).
So, my question is: which is the correct (if there is any) form to implement a gain-scheduling PID controller?
Thank you in advanced.
http://www.mathworks.com/help/simulink/slref/pidcontroller.html
Adolfo, it depends on the desired properties of your controller. The second choice allows you to make y(t) arbitrarily small by reducing the gains, whereas the first one yields y(t) = const after the gains have been gradually reduced to zero.
Hi Adolfo,
It is difficult to pinpoint either as "correct" or "incorrect" because it all depends on the way you make the gains time varying.
If it is a simple case of gain scheduling, I would say both are almost equivalent to each other.
If you wish to make the gains adaptive, then the equivalence depends on the specific adaptive technique you adopt, and the way you change the gains.
If you wish to make gains robust, the comparison changes again.
I would suggest some reading on adaptive PID control to start with (unless you have already done this, that is). There is a lot of stuff available on the web, both by way of IEEE and IET digital libraries.
With best wishes.
-Sanjay
Dear Adolfo,
In my oppinion, the second version seems to be more correct. Here the gain Ki(t) obeys its natural meaning, i.e., it multiplies the integral of the error.
Best regards,
Sergei
Dear Adolfo,
What I suggest is a supplement with the statements of colleagues : Using the improved PID nonlinear, or adaptive PID to solve the problem exposed.
Here is link :
-Design of an enhanced nonlinear PID controller
downloads.deusm.com/.../1477-Elsevier06.pdf
-2.153 Adaptive Control Lecture 7 Adaptive PID Control - MIT
aaclab.mit.edu/material/lect/lecture7.pdf
-Self-tuning of PID controllers by adaptive interaction - The ECE Home ...
www.ece.eng.wayne.edu/~flin/Conference/AI-PID.pdf
Best regards
Please refer to these articles:
https://www.researchgate.net/publication/313576646_From_PID_to_Nonlinear_State_Error_Feedback_Controller
https://www.researchgate.net/publication/312046377_Improved_Sliding_Mode_Nonlinear_Extended_State_Observer_based_Active_Disturbance_Rejection_Control_for_Uncertain_Systems_with_Unknown_Total_Disturbance
https://www.researchgate.net/publication/309655805_On_the_Improved_Nonlinear_Tracking_Differentiator_based_Nonlinear_PID_Controller_Design
Article From PID to Nonlinear State Error Feedback Controller
Article Improved Sliding Mode Nonlinear Extended State Observer base...
Article On the Improved Nonlinear Tracking Differentiator based Nonl...
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